Investigation into the relationship between acceleration and the angle of free fall downhill

Introduction

This is an investigation into the free fall of an object (in this case a small cart/trolley). When an object is placed on an inclined plane, the object is most likely to fall through the plane. Moreover, when the height of the plane is increased it was found that the object took less time to get to the other end. I decided to investigate the relationship between the angle of the slope of the plane and the acceleration of the object.

Aim

To investigate the extent of which the angle of the slope of the inclined plane is related to the acceleration of the object as it goes through the plane.

Independent Variables

Angle of the slope of the wooden ramp

Dependent Variables

Acceleration of the object

Controlled Variables

• Length of the plane
• Height of the plane
• Initial velocity of the object (= 0 ms-1)
• Distance travelled by the object

Hypothesis

The sine of the angle of the slope will be directly proportional to the acceleration of the object.

        From        

Where a is acceleration, θ is the angle of the slope, v is the final velocity, u is the initial velocity and t is time. For the second formula g is acceleration due to gravity (≈9.81 m s-2).

Equipment

• Wooden Ramp
• 1 Meter Ruler
• Stopwatch
• Bricks
• Books
• Dynamics Cart

Method of Data Collection

Procedure

1. A brick was set under one of the wooden ramp’s ends, leaving the other end touching the ground. The length (adjacent side to the angle)

0.176

0.646

0.638

0.689

0.661

0.632

0.653

0.029

The Table below shows the time taken for the cart to go downhill from the 5 experimental trials done for 5 different heights.

The uncertainty in the length and height of the plane was taken to be the half of the smallest non-zero unit on the ruler used for measurement. As a result, the uncertainty in the length and height of the plane is ±0.0005m.

It is to be noted that the time recorded used up to three decimal places, due to the use of a Casio stopwatch which differs from the usual stopwatch that would be used that would only have 2 decimal places.

The average time taken for the cart to go downhill for one of the heights was calculated from the raw data with the following mathematical formula:

The denominator was 5 because that was the number of trials experimented for one height. The average time was rounded up to 3 decimal places because the previous data had all been recorded with 3 decimal places.

The uncertainty in the average times is calculated by dividing the range of the data by 2, as shown in the following formula:

For example, for the uncertainty of the average time taken by the cart to go downhill with a height of 0.084 m:

The average time vs height was expressed in the following graph:

Processed Data

After obtaining results for the time taken for the cart to go down the slope, further calculations could be made, which include:

• Angle of inclination when the cart goes downhill
• Average Velocity of the cart, therefore acceleration of the cart too

To calculate the angle of the slope trigonometric functions were used, specifically the sine identity. The following formula was used

Therefore the angle is determined by

For example the sine and angle for the first height (0.084 m) would be as following

This calculation was applied to all the different height s and the results were expressed in the following graph, using 3 decimal places for all results:

The velocity was calculated taking into account the distance travelled and the time taken for the cart to go down the wooden ramp. Mathematically:

With this formula, we use as an example the set of trials for height number 1 (0.084 m):

The same calculation was done for the other heights; results were recorded and put into a table:

Improving the investigation

Measuring the heights, lengths and distances for this experiment seem to work fine, therefore they didn’t need much improvement. The main thing to improve for this experiment is the measurement of time.

The best way to eliminate the systematic error for time measuring would be to use better technology instead of just a stopwatch. This means that instead of a stopwatch, photo gates could’ve been used. Using photo gates would be better as it measures the exact point where the cart starts and finish and it can give you the value for velocity and acceleration immediately. However, the initial velocity should still be controlled carefully as the photo gate cannot control that.

Other than that, maybe a greater amount of trials could’ve been made or maybe the use of a third person with another stopwatch could help too. There are also some external factors that could’ve affected the results, for example the friction force of the ramp, maybe at times it could’ve affected the time the cart took to go downhill. This force, however, doesn’t have a major impact on the results, but for a future experiment it should be negligible in order to obtain more accurate results.

The final improvement would be to use a longer distance, using more than half a meter approximately for this experiment resulted in small results for the time. For a future experiment, using a longer distance (of at least twice as the once used for this experiment), would give a wider range of data and it would be helpful for further analysis and processing of data.